arXiv:physics/0308101v1 [physics.flu-dyn] 27 Aug 2003
Pis’ma v ZhETF
Weak turbulence of gravity waves.
A.I.Dyachenko+, A.O.Korotkevich+1), V.E.Zakharov+∗
+L.D. Landau Institute for Theoretical Physics RAS, 119334 Moscow, Russia
∗University of Arizona, Department of Mathematics, Tucson, USA
Submitted 16 April 2003
For the first time weak turbulent theory was demonstrated for the surface gravity waves. Direct numerical
simulation of the dynamical equations shows Kolmogorov turbulent spectra as predicted by analytical analysis
 from kinetic equation.
PACS: 47.11.+j, 47.27.-i, 47.27.Eq, 92.10.Hm
In this Letter we study numerically the steady
Kolmogorov spectra for spatially homogeneous gravity
waves. According to the theory of weak turbulence
the main physical process here is the stationary en-
ergy flow to the small scales, where the energy dissi-
pates [1, 2]. This flow is described by kinetic equation
which has power-like solutions – Kolmogorov spectra.
This straightforward picture takes place experimentally
and numerically for different physical situations. For
capillary waves it was observed on the surface of liq-
uid hydrogen , . The numerical simulation of this
process was performed in . In nonlinear fiber optics
these spectra were demonstrated in numerical simula-
tion .There are many other results [7, 8, 9, 10, 11].
One of the most interesting applications of the weak
turbulence theory is the surface gravity waves. From
the pioneering article by Toba  to the most recent
observations  many experimentalists get the spectra
predicted by the weak turbulence theory. But these ex-
periments cannot be treated as a complete confirmation
because the Zakharov-Filonenko spectrum is isotropic,
while observed spectra are essentially anisotropic. It is
worth to say that the wave kinetic equation, which is
the keystone of this theory, was derived under several
assumptions. Namely, it was assumed, that the phases
of all interacting waves are random and are in state of
chaotic motion. The validity of this proposition is not
clear a priori. The direct numerical simulation of non-
linear dynamical equations can give us a confirmation
is this assumption valid or not. But for particular case
of gravity surface waves the numerical confirmation was
absent in spite of significant efforts were applied. The
only successful attempt in this direction was the sim-
ulation of freely decaying waves . The reason for
that for our opinion was concerned with a choice of
numerical scheme parameters. Namely, the numerical
simulation is very sensitive to the width of resonance
of four-waves interaction. It must be wide enough to
provide resonance on the discrete grid, as it was studied
in  for decay of the monochromatic capillary wave.
From the other hand it has to be not too wide (due to
nonlinear frequency shift) when the weak turbulent con-
ditions fail. We have spent significant efforts to secure
the right choice of numerical parameters. As a result we
have obtained the first evidence of the weak turbulent
Kolmogorov spectrum for energy flow for surface grav-
ity waves. The numerical simulation was surprisingly
time consuming (in comparison to capillary waves tur-
bulence), but finally we clearly get spectrum for surface
which is in the agreement with real experiments [12, 13].
Theoretical background. — Let us consider the po-
tential flow of an ideal incompressible fluid of infinite
depth and with a free surface. We use standard nota-
tions for velocity potential φ(r,z,t),r = (x,y);v = ∇φ
and surface elevation η(r,t). Fluid flow is irrotational
△φ = 0. The total energy of the system can be repre-
sented in the following form
H = T + U,
where g – is the gravity acceleration. It was shown 
that under these assumptions the fluid is a Hamiltonian
A.I.Dyachenko, A.O.Korotkevich, V.E.Zakharov
where ψ = φ(r,η(r,t),t) is a velocity potential on the
surface of the fluid. In order to calculate the value of
ψ we have to solve the Laplas equation in the domain
with varying surface η. This problem is difficult. One
can simplify the situation, using the expansion of the
Hamiltonian in powers of ”steepness”
?ˆk(η(ˆkψ)) + η△ψ
For gravity waves it is enough to take into account terms
up to the fourth order. Hereˆk is the linear operator
corresponding to multiplying of Fourier harmonics by
modulus of the wavenumber k. In this case dynamical
equations (4) acquire the following form
ˆkψ − (∇(η∇ψ)) −ˆk[ηˆkψ]+
−[ˆkψ]ˆk[ηˆkψ] − [ηˆkψ]△ψ + Dr+ Fr.
Here Dris some artificial damping term used to provide
dissipation at small scales; Fris a pumping term corre-
sponding to external force (having in mind wind blow,
for example). Let us introduce Fourier transform
With these variables the Hamiltonian (5) acquires the
H = H0+ H1+ H2+ ...,
×δ(k1+ k2+ k3)dk1dk2dk3,
×δ(k1+ k2+ k3+ k4)dk1dk2dk3dk4,
Lk1k2= (k1k2) + |k1||k2|,
+|k2+ k3| + |k2+ k4|) − |k1| − |k2|].
It is convenient to introduce the canonical variables ak
as shown below
2(|k1+ k3| + |k1+ k4|+(8)
this is the dispersion relation for the case of infinite
depth. The similar formulas can be derived in the case
of finite depth . With these variables the equations
(4) take the following form
˙ ak= −iδH
The dispersion relation (10) is of the ”non-decay type”
and the equations
ωk1= ωk2+ ωk3,
k1= k2+ k3
have no real solution.
small nonlinearity, the cubic terms in the Hamiltonian
can be excluded by a proper canonical transformation
a(k,t) −→ b(k,t) . The formula of this transforma-
tion is rather bulky and well known [17, 18], so let us
omit the details here.
For statistical description of a stochastic wave field
one can use a pair correlation function
It means that in the limit of
k′ >= nkδ(k − k′).(13)
The nkis measurable quantity, connected directly with
observable correlation functions. For instance, from (9)
one can get
In the case of gravity waves it is convenient to use an-
other correlation function
k′ >= Nkδ(k − k′).(15)
The function Nkcannot be measured directly. The rela-
tion connecting nkand Nkis rather complex in the case
of fluid of finite depth. But in the case of deep water it
becomes very simple 
where µ = (ka)2, here a is a characteristic elevation
of the free surface. In the case of the weak turbulence
µ << 1. The correlation function Nk obey the kinetic
= st(N,N,N) + fp(k) − fd(k),(17)
st(N,N,N) = 4π
×(Nk1Nk2Nk3+ NkNk2Nk3− NkNk1Nk2−
−NkNk1Nk3)δ(k + k1− k2− k3)dk1dk2dk3.
Weak turbulence ...
The complete form of matrix element Tk,k1,k2,k3can be
found in many sources [1, 2, 17]. Function fp(k) in (17)
corresponds to wave pumping due to wind blow for ex-
ample. Usually it is located on long scales. Function
fd(k) represents the absorption of waves due to viscos-
ity and wave-breaking. None of this functions are known
to a sufficient degree.
Let us consider stationary solutions of the equation
(17) assuming that
• The medium is isotropic with respect to rotations;
• Dispersion relation is a power-like function ω =
Under this assumptions one can get Kolmogorov solu-
Here d is a spatial dimension (d = 2 in our case).
The first one is a Kolmogorov spectrum, correspond-
ing to a constant flux of energy P to the region of small
scales (direct cascade of energy). The second one is Kol-
mogorov spectrum, describing inverse cascade of wave
action to large scales, and Q is a flux of action. In both
cases C1and C2are dimensionless ”Kolmogorov’s con-
In the case of deep water ω =√gk and, apparently,
β = 3. It is known since  that on deep water
In the same way  for second spectrum
In this Letter we will explore the first spectrum (en-
ergy cascade). Using (14) one can get
Numerical Simulation — Dynamical equations (6)
are very hard for analytical analysis. One of the main
obstacles is theˆk-operator which is nonlocal. However,
using Fourier technique practically makes no difference
between derivative andˆk. The numerical simulation of
the system is based upon consequent application of fast
Fourier transform algorithm. The details of this numer-
ical scheme will be published separately.
For numerical integration of (6) we used the func-
tions F and D defined in Fourier space
fk= 4F0(k − kp1)(kp2− k)
γk= −γ1,k ≤ kp1,
γk= −γ2(k − kd)2,k > kd.
Here Rk(t) is the uniformly distributed random num-
ber in the interval (0,2π). We have solved system of
equations (6) in the periodic domain 2π×2π (the wave-
numbers kxand kyare integers in this case). The size
of the grid was chosen 256 × 256 points. Gravity accel-
eration g = 1. Parameters of the damping and pumping
were the following: kp1= 5, kp2= 10, kd= 64. Thus
the inertial interval is about half of decade.
During the simulations we paid special attention
to the problems which could ”damage” the calcula-
tions. First of all, the ”bottle neck” phenomenon at the
boundary between inertial interval and dissipation re-
gion. This effect is very fast, but can be effectively sup-
pressed by proper choice of damping value γ2in the case
of moderate pumping values F0. The second problem is
the accumulation of ”condensate” in low wave numbers.
This mechanism for the case of capillary waves was ex-
amined in details in . This obstacle can be over-
come by simple adaptive damping scheme in the small
wave numbers. After some time system reaches the sta-
tionary state, where the equilibrium between pumping
and damping takes place. Important parameter in this
state is the ratio of nonlinear energy to the linear one
For example, in the case of F0 = 2 × 10−4,γ1 =
1 × 10−3,γ2 = 400 the level of nonlinearity was equal
to (H1+ H2)/H0 ≃ 2 × 10−3. The Hamiltonian as a
function of time is shown in Fig. 1.
The surface elevation correlator function appears to
be power-like in the essential part of inertial interval,
where the influence of pumping and damping was small.
The correlator is shown in Fig. 2.
One can try to estimate the exponent of the spec-
trum. It is worth to say that an alternative spectrum
was proposed earlier by Phillips . That power-like
spectrum is due to wave breaking mechanism and gives
us a surface elevation correlator as Ik∼ k−4. Compen-
sated spectra are shown in the Fig. 3. It seems to be
an evidence, that the Kolmogorov spectrum predicted
by weak turbulence theory better fit the results of the
A.I.Dyachenko, A.O.Korotkevich, V.E.Zakharov
0500010000 1500020000 250003000035000
Fig.1. Hamiltonian as a function of time.
1 10 100
Fig.2. The logarithm of the correlator function of sur-
face elevation as a function of logarithm of the wave
The inertial interval was rather narrow (half a
decade). But the obtained results allow us to conclude,
that accuracy of experiment was good enough under the
time constraints of simulation (we get the steady state
after 20-30 h using available hardware, and we need sev-
eral days to average |ηk|2function). The simulation on
larger grid (512 × 512, for example) can make the ac-
curacy better. But even these results give us a clear
This work was supported by RFBR grant 03-01-
00289, INTAS grant 00-292, the Programme “Nonlinear
dynamics and solitons” from the RAS Presidium and
“Leading Scientific Schools of Russia” grant, also by US
Army Corps of Engineers, RDT&E Programm, Grant
DACA 42-00-C0044 and by NSF Grant NDMS0072803.
Fig.3. Compensated correlators in inertial interval for
different values of the compensation power: z = 3.5
solid line (weak turbulence theory), z = 4.0 dashed line
Also authors want to thank creators of the open-
source fast Fourier transform library FFTW  for this
fast, portable and completely free piece of software.
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